This paper studies uncertainty and its effect on system response displacement. The paper also describes how IT2MFs (interval type-2 membership functions) differentiate from T1MFs (type-1 membership functions) by adding uncertainty. The effect of uncertainty is modeled clearly by introducing a technique that describes how uncertainty causes membership degree reduction and changing the fuzzy word meanings in fuzzy logic controllers (FLCs). Several criteria are discussed for the measurement of the imbalance rate of internal uncertainty and its effect on system behavior. Uncertainty removal is introduced to observe the effect of uncertainty on the output. The theorem of uncertainty avoidance is presented for describing the role of uncertainty in interval type-2 fuzzy systems (IT2FSs). Another objective of this paper is to derive a novel uncertainty measure for IT2MFs with lower complexity and clearer presentation. Finally, for proving the affectivity of novel interpretation of uncertainty in IT2FSs, several investigations are done.

Type-n fuzzy sets were discussed generally and comprehensively in [

Interval type-2 fuzzy sets have widely been accepted as more capable of modeling higher orders of uncertainty than type-1 fuzzy sets [

The authors presented an ensemble of fuzzy systems in [

The concept of uncertainty in fuzzy systems is interpreted in a new manner as illustrated in Figure

The role of uncertainty in an IT2FS or IT2 FLC.

Uncertainty removal is introduced to observe the effect of uncertainty on the output. In Section

The direct role of uncertainty is introduced as the main difference between T1FSs and T2FSs. The justification of application of IT2FLSs in fuzzy systems is highly dependent on uncertainty issue. This paper aims to recognize different types of uncertainty by introducing new definitions and expressions, thus creating informed and goal-oriented maneuvers during the design process and increasing the application of interval T2 fuzzy controllers.

Another objective of this paper is about new uncertainty measure for interval type-2 fuzzy membership functions (IT2MFs). Uncertainty measure is a necessity because to use fuzzy sets (FSs) as granules in general theory of uncertainty (GTU), which is introduced by Zadeh [

Klir and Folger [

“The principle of minimum uncertainty, which states that solutions with the least loss of information should be selected, can be used in simplification and conflict resolution problems.”

“The principle of maximum uncertainty, which states that a conclusion should maximize the relevant uncertainty within constraints given by the verified premises, is widely used within classical probability framework [

“The principle of uncertainty invariance, which states that the amount of uncertainty should be preserved in each transformation of uncertainty from one mathematical framework to another, is widely studied in the context of probability-possibility transformation [

This paper tries to discuss the effect of uncertainty of IT2FS based on the first principal of uncertainty.

The remainder of this paper is organized as follows. Section

Zadeh points out in [

Fuzzy systems are mainly applied for calculations that use lexical variables (i.e., CW) [

Principal function and one of its uncertaintified functions.

Four sources with different nature were mentioned for uncertainty in [

If we divide the uncertainty sources introduced in [

Basic IT2MF shapes.

In a specific problem, choosing the same type of MF for all inputs and outputs is preferred.

Regardless of whether the triangular, trapezoidal, or Gaussian MFs are selected at points where the input range has quantitatively higher or lower membership degrees, the uncertainty band will be narrower and wider at points that have medium membership degrees, respectively. This model is consistent with our attitudes toward words. For instance, at a low mode, the membership degree is high when the uncertainty of placing very small data in a field becomes low (Points zero and one in low shape in Figure

Another type of uncertainty is related to data. Two out of 4 items discussed in [

If the uncertainty in the nature of a word is applied to a T1MF, we will achieve the same common IT2MFs. However, if the same approach is implemented on data uncertainty, we will not achieve IT2MFs with common forms. In this example, two types of uncertainties are applied to a basic T1 fuzzy function in two steps.

Sample data that denotes uncertainty as an interval.

Interval | [2.7, 3.3] | [0.7, 3.3] | [−1.3, 3.3] | [−3.3, 3.3] |

Average | 3 | 2 | 1 | 0 |

| ||||

Interval | [6.3, 7.7] | [5.4, 6.6] | [4.5, 5.5] | [3.6, 4.4] |

Average | 7 | 6 | 5 | 4 |

| ||||

Interval | [11, 11] | [9.825, 0.175] | [8.65, 9.23] | [7.475, 8.525] |

Average | 11 | 10 | 9 | 8 |

In this example, we assume that the general field of the data related is the verbal expression of “low,” which is defined as T1 fuzzy MF (Figure

(a) Uncertain space of measured data. (b) Application of uncertain space to one TIMF. (c) Obtained IT2MF.

Each set of optional data, which represents one set of figures in each interval of uncertainty in Table

In the following step, we intend to inject data uncertainty to an IT2MF that defines only the natural uncertainty inside the verbal expression (Figure

(a) IT2MF based on the intrinsic description of uncertainty. (b) Application of uncertain space to one type-2 MF. (c) Obtained IT2MF.

According to the definition provided in this section, the MFs of Figures

The effect of these different MFs on system behavior is investigated in Section

Discussions have been provided in [

For each IT2MF, one defines a function of

(a) Typical IT2MF and its principal function. (b) Uncertain bandwidth function

The absolute ratio of the surface areas on both sides of a line to the length of the COG of the principle function enclosed between the upper and lower bounds of uncertainty band is called the difference index in internal uncertainty, also referred to as the absolute ratio of surface (ARS). If the uncertainty bandwidth by

The role of ARS, which is the role of the surface area between two uncertainty bands in internal uncertainty in [

For example, as shown in Figure

(a) Both sides are certain; (b) left side is slightly uncertaintified; (c) left side is more uncertaintified; (d) addition of uncertainty in the right side of (c).

IT2MF that indicates the upper bound;

It is concluded that

Figure

The amount of uncertainty added to the right and left sides of

Theorem of Uncertainty Avoidance points out that “The response of system avoids uncertainty.” In case of IT2FMs, it means that the center of gravity of uncertaintified membership functions is displaced toward the less uncertaintified domain or, in other words, toward the more clarified domain. The more the amount of imbalance of the created (or available) uncertainty on both sides of the center of gravity of principle membership function, the more the displacement will tend toward certainty. In other words, if some uncertainty is injected to one side of the center of gravity of IT2MF, the center of gravity will be shifted toward the other side.

In this paper, we measure the rate of uncertainty based on the power of total uncertainty to push the response of system to the opposite side considering the aforementioned theorem.

In this method, we add a completely certain membership function (with membership degree is equal to one) to IT2MF in the right hand in such a way that the COG of new established principal function is positioning in the conjunction point of the added part with earlier IT2MF.

Considering the aforementioned theorem, the COG of new established IT2MF must be displaced to the right side in (Figure

(a) A typical IT2MF. (b) Extended principal function and extended IT2MF.

In Method

The domain of discourse of shown IT2MF in Figure

The principal function must be extended to the right side by adding a T1 membership function with fixed and certain membership degree of “one.” We lengthen the added part to the right in Figure

For obtaining “

Using KM algorithm, we calculate the COG of extended IT2MF.

The direct distance between

In this method, we use an approximation for calculating the COG of IT2MF during extracting a criterion for measuring uncertainty.

In this method, we extend the IT2MF as described in part 2 of Method

Given that switching point in KM algorithm is positioned in the right side of

Given that switching point in KM algorithm is positioned in the right side of

The COG of extended principal function is shown in

Extended IT2MF based on Method

According to Theorem of Uncertainty Avoidance, if uncertainty increases in one side of COG of principal function, the new COG moves to the other side. In other words, we can see this effect similar to the situation in which the membership grades of principal function are decreased in that side that uncertainty increased. Equation (

Considering the definition of ARS in (

In the general condition of the proposed method, the response of IT2FS is not always completely in accord with COG. We propose a technique that considers this difference (refer to Table

Basic IT2FS words: defuzzified output obtained by using the proposed method and difference percentage compared to the COG calculated by the KM algorithm.

UMF | LMF | KM | Proposed | ARS | % E | |
---|---|---|---|---|---|---|

(1) | [0, 0, 0.14, 1.97, 1] | [0, 0, 0.05, 0.66, 1] | 0.47 | 0.49 | 0.27 | 1.0 |

(2) | [0, 0, 0.14, 1.97, 1] | [0, 0, 0.01, 0.13, 1] | 0.56 | 0.61 | 0.96 | 2.5 |

(3) | [0, 0, 0.26, 2.63, 1] | [0, 0, 0.05, 0.63, 1] | 0.63 | 0.66 | 0.47 | 1.1 |

(4) | [0, 0, 0.36, 2.63, 1] | [0, 0, 0.05, 0.63, 1] | 0.64 | 0.67 | 0.49 | 1.1 |

(5) | [0, 0, 0.64, 2.47, 1] | [0, 0, 0.10, 1.16, 1] | 0.66 | 0.66 | 0.24 | 0.0 |

(6) | [0, 0, 0.64, 2.63, 1] | [0, 0, 0.09, 0.99, 1] | 0.67 | 0.68 | 0.29 | 0.3 |

(7) | [0.59, 1.50, 2.00, 3.41, 1] | [0.79, 1.68, 1.68, 2.21, 0.74] | 1.75 | 1.74 | 0.45 | 0.3 |

(8) | [0.38, 1.50, 2.50, 4.62, 1] | [1.09, 1.83, 1.83, 2.21, 0.53] | 2.13 | 2.11 | 0.75 | 0.5 |

(9) | [0.09, 1.25, 2.50, 4.62, 1] | [1.67, 1.92, 1.92, 2.21, 0.30] | 2.19 | 2.28 | 0.99 | 0.2 |

(10) | [0.09, 1.50, 3.00, 4.62, 1] | [1.79, 2.28, 2.28, 2.81, 0.40] | 2.32 | 2.33 | 0.91 | 0.2 |

(11) | [0.59, 2.00, 3.25, 4.41, 1] | [2.29, 2.70, 2.70, 3.21, 0.42] | 2.59 | 2.59 | 0.93 | 0.0 |

(12) | [0.38, 2.50, 5.00, 7.83, 1] | [2.88, 3.61, 3.61, 4.21, 0.35] | 3.90 | 3.94 | 0.93 | 0.5 |

(13) | [1.17, 3.50, 5.50, 7.83, 1] | [4.09, 4.65, 4.65, 5.41, 0.40] | 4.56 | 4.57 | 0.95 | 0.1 |

(14) | [2.59, 4.00, 5.50, 7.62, 1] | [4.29, 4.75, 4.75, 5.21, 0.38] | 4.95 | 4.98 | 0.89 | 0.6 |

(15) | [2.17, 4.25, 6.00, 7.83, 1] | [4.79, 5.29, 5.29, 6.02, 0.41] | 5.13 | 5.13 | 0.98 | 0.0 |

(16) | [3.59, 4.75, 5.50, 6.91, 1] | [4.86, 5.03, 5.03, 5.14, 0.27] | 5.19 | 5.21 | 0.90 | 0.6 |

(17) | [3.59, 4.75, 6.00, 7.41, 1] | [4.79, 5.30, 5.30, 5.71, 0.42] | 5.41 | 5.41 | 0.99 | 0.0 |

(18) | [3.38, 5.50, 7.50, 9.62, 1] | [5.79, 6.50, 6.50, 7.21, 0.41] | 6.50 | 6.50 | 0.82 | 0.0 |

(19) | [4.38, 6.50, 8.00, 9.41, 1] | [6.79, 7.38, 7.38, 8.21, 0.49] | 7.16 | 7.15 | 0.82 | 0.2 |

(20) | [4.38, 6.50, 8.00, 9.41, 1] | [6.79, 7.38, 7.38, 8.21, 0.49] | 7.16 | 7.15 | 0.90 | 0.2 |

(21) | [4.38, 6.50, 8.25, 9.62, 1] | [7.19, 7.58, 7.58, 8.21, 0.37] | 7.25 | 7.21 | 0.86 | 0.7 |

(22) | [5.38, 7.50, 8.75, 9.81, 1] | [7.79, 8.22, 8.22, 8.81, 0.45] | 7.90 | 7.87 | 0.86 | 0.6 |

(23) | [5.38, 7.50, 8.75, 9.83, 1] | [7.69, 8.19, 8.19, 8.81, 0.47] | 7.91 | 7.88 | 0.45 | 0.6 |

(24) | [5.38, 7.50, 8.75, 9.81, 1] | [7.79, 8.30, 8.30, 9.21, 0.53] | 8.01 | 8.01 | 0.65 | 0.0 |

(25) | [5.38, 7.50, 9.00, 9.81, 1] | [8.29, 8.56, 8.56, 9.21, 0.38] | 8.03 | 7.97 | 0.90 | 1.3 |

(26) | [5.98, 7.75, 8.60, 9.52, 1] | [8.03, 8.36, 8.36, 9.17, 0.57] | 8.12 | 8.12 | 0.65 | 0.0 |

(27) | [7.37, 9.41, 10, 10, 1] | [8.72, 9.91, 10, 10, 1] | 9.30 | 9.31 | 0.24 | 0.3 |

(28) | [7.37, 9.82, 10, 10, 1] | [9.74, 9.98, 10, 10, 1] | 9.31 | 9.23 | 0.28 | 3.0 |

(29) | [7.37, 9.59, 10, 10, 1] | [8.95, 9.93, 10, 10, 1] | 9.34 | 9.35 | 0.28 | 0.3 |

(30) | [7.37, 9.73, 10, 10, 1] | [9.34, 9.95, 10, 10, 1] | 9.37 | 9.34 | 0.47 | 1.1 |

(31) | [7.37, 9.82, 10, 10, 1] | [9.37, 9.95, 10, 10, 1] | 9.38 | 9.34 | 0.48 | 1.5 |

(32) | [8.68, 9.91, 10, 10, 1] | [9.61, 9.97, 10, 10, 1] | 9.69 | 9.67 | 0.36 | 1.5 |

Difference (error) RMS% compared to KM 0.95%.

The following results are obtained from the many calculations and simulations conducted by the authors of this paper to extract the closed formula of COG and propose simple and effective formulas.

According to the aforementioned equations,

Calculations and simulations at various different conditions lead to more complex results. The COG can be shifted by the proposed formula and by calculating the COG by using the KM algorithm. However, one approach is slower or faster than the other approach.

According to the results obtained from the simulation and calculation in which IT2MF is more asymmetric but has equal uncertainty areas in both sides of the COG of the principal function, (

This method has a high degree of freedom (

Our method eases the defuzzifying of IT2MF to obtain accurate results on the main feature of IT2FSs. In case of slight output differences between our method and the KM algorithm, no mathematic proof exists that shows that the outputs gained by KM algorithm are better than our method. On the contrary, the existence of concepts and reasons behind our proposed method provides a designer with more opportunities to manage parameters related to uncertainty in IT2 fuzzy controllers in engineering and industrial affairs conveniently.

Investigation 1 examines the affectivity of the proposed uncertainty measures. Investigation 2 shows the illustrative concept of the effect of uncertainty on membership degrees. Investigation 3 shows the comparative outputs as a result of applying the proposed formula and KM algorithm of 32 basic IT2MF words. A single-input single-output (SISO) fuzzy system is introduced in “Investigation 4” to compare clearly the outputs created in different uncertainty imbalance situations and by different methods. Investigation 5 discusses on a comparison between collapsing method [

The common form of the trapezoidal and triangular MF can be described by a five-number vector [

Common form of trapezoidal and triangle MFs.

Two IT2MFs with equal domains and different uncertainty bands.

According to (

Two uncertainty measures have been shown in Figure

COG of extended IT2MF and position of

(a) IT2MF and its principal and output-based imbalanced uncertainty; (b) uncertain bandwidth function; (c) COGs of principal, output-based uncertainty imbalances, and IT2MF (KM) functions.

By using (

The COG of IT2MF in Figure

Here,

Calculate the output by using (

The output is affected by various uncertainties separately (Figure

Effect of uncertainty on outputs of SISO (Investigation 4).

the MF of the upper and lower bands that are 0.8 and 0.2 in the

symmetric triangle MF, UMF =

asymmetric Gaussian MF, UMF =

In all the above conditions, calculate the output by using (

The output is affected by various uncertainties separately (Figure

Effect of uncertainty on outputs of a simple fuzzy controller (Investigation 6).

However, we considered the effect of uncertainty in Figure

Our method eased the defuzzifying of IT2MF to obtain accurate results on the feature of IT2FSs. In case of slight output differences between our method and the KM algorithm, no mathematical proof exists to show that the outputs gained by KM algorithm are better than our method. On the contrary, the existence of concepts and reasons behind our proposed method provides designers with more opportunities to manage parameters related to uncertainty in T2 fuzzy systems in engineering and industrial affairs conveniently.

Designers who work with fuzzy systems must have information on the behavior of the fuzzifier, inference, and defuzzifier methods used in the system. No defuzzification method exists that is suitable in all systems and all conditions. If method A produces better results than method B, B is better than A if conditions or systems change. The experience of a designer plays a major role in the selection of an appropriate method. In our case, understanding and absorbing system behavior is easier.

This paper presented the Theorem of Uncertainty Avoidance and used it for uncertainty measuring in IT2FMs. The proposed methods provide simple closed formulas for calculating total uncertainty of a membership function. This paper brought up a new vision to the problem of uncertainty measure. The measurement is based on the power of uncertainty to push the COG of principal function to a completely certain domain. The results of this paper provide a new perspective on the relationship between uncertainty and fuzzy system output. For each T1MF, uncertaintified functions are presented to be more complete than common IT2MFs. In addition, we show that uncertainty reduces the value of membership degrees and the absolute value of words. Higher uncertainty causes a higher reduction of values. For example, if the input and output of a system contain “low,” “medium,” and “large” words, after injecting uncertainty to inputs, “low” will shift to “medium,” “medium” to “large,” and “large” to “medium.” Results show that the uncertainty reduces the value of the membership degree proportionally. The concept of words is also shifted toward the opposite neighbor words in the system output by the uncertainty of system inputs proportionally. The proposed technique for uncertainty removing can be considered as a closed formula for calculating the COG of IT2MF with acceptable accuracy. On the contrary, the existence of concepts and reasons behind the new interpretation of uncertainty provides designers more opportunities to manage parameters related to uncertainty in interval T2 fuzzy controllers in engineering and industrial affairs conveniently.

The authors would like to acknowledge the support of the Universiti Sains Malaysia fellowship.